"First off" vs "first"

1. First off we need to write down a word; second we need...
2. First we need to write down a word; second we need...

What's the subtle difference between "first off" and "first"?

Moreover, what's the exact meaning of the word "off" here?

Solution 1:

First off is an idiom. As referenced here, it's an "Extragrammatical" Form...

Sometimes even the form itself is not predictable from the grammar. Phrasal idioms of this sort are what are called in the Fillmore/Kay/O'Connor article extragrammatical idioms [reference]. Examples: "by and large" "first off", "all of a sudden". The interpreter could not even know, without learning them separately, that these expressions are sayable in English. The grammar of English has no mechanism for conjoing a preposition ("by") with an adjective ("large").

What that means is there's no way for us to infer what exactly the word off is doing there, but my money is on this suggestion...

It's a idiom and might be short for "first off the block" (sprint athlete's starting block).

As regards any "subtle difference", I don't think there's any difference in meaning. It's just that the idiomatic form is more informal/colloquial. Some may say it's "ungrammatical", but that's fairly pointless given the above explanation of what an "extragrammatical idiom" actually means.

Solution 2:

The difference is that first off is colloquial, and first alone isn’t. The earliest citation in the OED is from Mark Twain in 1880: ‘First-off, I thought it would certainly give me the botts.’

The addition of the particle may be thought necessary in speech to strengthen what would otherwise be a monosyllable and thus fail to attract the listener’s attention. Up is similarly used in ‘first up’.

Solution 3:

First off is more colloquial. First is used for eg a formal set of instructions.

Solution 4:

The only practical reason to favour first off is that first has more uses, and fewer syllables.

That first has more uses means that when we say first off it's clearer that we mean only in the sense that we are going to get on to second and subsequent items shortly.

That first off has one more syllable has a practical advantage in spoken English, because when we list items as "First, …. Second, …." or "Firstly, …. Secondly…." then we often want to stress that we are only detailing part one of a multi-part plan or explanation. It's easier to stress across two stressed syllables than on either one, or one where the second syllable has less stress.

For this reason, we can see the practical need it serves that makes it popular despite being considered informal or even colloquial.